Zener resonance in a dynamic Wannier-Stark ladder: Two miniband model

Quasienergy structures of Floquet states in strongly biased superlattices (with a static electric field of $F_0$) that are further driven by a sinusoidal electric field (with an amplitude of $F_1$) are calculated in a two miniband model when Zener resonance between the two minibands is significant. It is found that the quasienergies are affected pronouncedly by static and dynamic Zener tunnelings pertinent to $F_0$ and $F_1$, respectively, where both effects simultaneously couple Wannier-Stark ladder (WSL) subband states that are energetically aligned with each other. The dynamic Zener tunneling causes two lobes of quasienergy parent bands, which are ascribable to the different superlattice minibands and almost degenerate at $F_1=0$, to swerve sharply with increasing $F_1$. As $F_1$ becomes much larger, due exclusively to the static Zener tunneling, each split band undergoes a strong anticrossing with another lobe pertaining to the adjacent photon sideband. Furthermore, due mostly to the dynamic Zener tunneling, tendency toward band collapse or bandwidth narrowing characteristic of the usual dynamic WSL based on the single miniband picture is not observed here any longer for large $F_0$. On the contrary, bandwidth minima of one of the two split parent bands alternate with those of the other, and hence bandwidth narrowing does not occur simultaneously in every lobe at a single $F_1$.

Figure 1

A schematic diagram of the geometry of the DWSL under study. The time-dependent electric field $F_0+F_1\cos \omega t$ is applied along the crystal growth direction. The resonant transition mediated by $\omega$ is depicted. For the meanings of the terms, PAT, static ZT, and dynamic ZT, see the text.

Figure 2

The WSL energies of ${\mathcal{E}}_{l\left(b\right)}$ as a function of $F_0$. Here, some WSL subband states are explicitly labeled for the sake of clarity. The anticrossings of the state 0(2) formed with the state $-3\left(1\right)$ at $F_0=24\mathrm{kV}∕\mathrm{cm}$ and with the state $-1\left(1\right)$ at $F_0=72\mathrm{kV}∕\mathrm{cm}$ are specified by dashed circles. Both anticrossings are discussed in detail in the text,

Figure 3

The wave functions ${\varphi }_j\left(z\right)$ of the WSL subband state $j$ associated with the anticrossings marked in Fig. 2 as a function of $z$. The panel (a) for the states of $j=-3\left(1\right)$ and 0(2) at $F_0=24\mathrm{kV}∕\mathrm{cm}$, and the panel (b) for the states of $j=-1\left(1\right)$ and 0(2) at $F_0=72\mathrm{kV}∕\mathrm{cm}$. The wave functions represented by solid lines are obtained by use of $N_b=2$, and those by dotted and chain lines are by use of $N_b=1$ and 6, respectively, for the purpose of comparison. The geometries of the present WSLs are also depicted.

Figure 4

The quasienergies $E$ of the DWSL with (a) $F_0=24\mathrm{kV}∕\mathrm{cm}$ and (b) $F_0=72\mathrm{kV}∕\mathrm{cm}$ as a function of $F_1$. The results are obtained with (i) the full effects of the PAT, the dynamic ZT, and the static ZT, (ii) the effects of the PAT and the dynamic ZT, (iii) the effects of the PAT and the static ZT, and (iv) the effect of the PAT only. In every panel, the quasienergy lobes pertaining to the parent bands are depicted in the middle of the ordinate. The indices of the parent bands are indicated only in the panel of (iv). Dashed lines represent numerical inaccuracy incurred by the finite number of $N_z$, namely, the surface effect, and by poor convergence of numerical diagonalizations due to the limited number of $N_{\mathrm{ph}}$, which stands out in the large $F_1$ region of the panel (iii).